Title Zygmund Type Estimates and Mapping Properties of Operatorswith Power-Logarithmic Kernels in Generalized Holder Spaces

Author(s) Kilbas, Anatoly A.; Saigo, Megumi; Bubakar, Silla

Citation 数理解析研究所講究録 (1993), 821: 87-100

Issue Date 1993-02

URL http://hdl.handle.net/2433/83191

Right

Type Departmental Bulletin Paper

Textversion publisher

Kyoto University

87

Zygmund Type Estimates and Mapping Properties ofOperators with Power-Logarithmic Kernels

in Generalized H\"older Spaces

Anatoly A. Kilbas* (ベラルーシ国立大学)

Megumi Saigo\dagger [西郷 恵] (福岡大学理学部)

Silla Bubakar* (ベラルーシ国立大学)

Abstract

Zygmund type estimates for the integral with power-logarithmic kernel with a vari-able upper limit and for its inversion are obtained. The results are applied to studymapping properties of operators with power-logarithmic kernels in generalized II\"olderspaces $H_{0}^{\omega}([a, b])$ with any modulus of continuity $\omega$ and to prove an isomorphism be-tween these spaces realized by the above operators.

1. Introduction

Let $H_{0^{\lambda}}([a, b])$ be the space $H^{\lambda}([a, b])$ of H\"olderean functions $f$ on a finite interval $[a, b]$ ofthe real axis such that $f(a)=0$. It is well known by a classical Hardy-Littlewood theorem[3] that if $0<\lambda<1,0<\alpha<1$ and $\lambda+\alpha<1$ , the Riemann-Liouville fractional integrationoperator

(1) $(I_{a+}^{\alpha} \phi)(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}\phi(t)dt$

maps the H\"older space $H_{0}^{\lambda}([a, b])$ boundedly into $H_{0}^{\lambda+\alpha}([a, b])$ . This statement was gen-eralized in various directions (see [12, \S \S 3,4,13,17] for historical notes and the review ofsuch results). In particular, in [11] and [9] the Hardy-Littlewood theorem was extended tothe weighted H\"older spaces $H_{0^{\lambda}}([a, b];\rho)=\{g : \rho g\in H_{0^{\lambda}}([a, b])\}$ with the power weights$\rho(x)=(x-a)^{\mu}(b-x)^{\nu}$ and

(2) $\rho(x)=\prod_{k=1}^{m}|x-x_{k}|^{\mu k},$ $a\leqq x_{1}\leqq\ldots\leqq x_{m}\leqq b$ ,

*Department of Mathematics and Mechanics, Byelorussian State University, Minsk 220080, Belarusi\dagger Department of Applied Mathematics, Fukuoka University, Fukuoka 814-01, Japan

数理解析研究所講究録第 821巻 1993年 87-100

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concentrated at the end and inner points of $[a, b]$ , respectively. Moreover, in [9] it was shownthat $I_{a+}^{\alpha}$ implements an isomorphism between the spaces $H_{0}^{\lambda}([a, b];\rho)$ and $H_{0}^{\lambda+\alpha}([a, b];\rho)$ .We also note that such an isomorphism between the spaces $H_{0^{\lambda}}([a, b])$ and $H_{0}^{\lambda+\alpha}([a, b])$ wascontained in embryo in [3].

Under certain conditions on the characteristic $\omega$ in [7] (see $al$so [12, \S 13.6] and [14]) it wasproved that the operator $I_{a+}^{\alpha}$ of fractional integration implements an isomorphism betweenthe generalized H\"older spaces $H_{0}^{\omega}([a, b])$ and $H_{0}^{\omega_{\alpha}}([a, b])$ with $\omega_{\alpha}(h)=h^{a}\omega(h)$ . This resultwas extended in [8], [13] and [14] to the weighted generalized H\"older spaces $H_{0}^{\omega}([a, b];\rho)$

with $\rho(x)=(x-a)^{\mu}(b-x)^{\nu}$ , in [4] to the generalized H\"older spaces $H_{p}^{\omega}$ with the integralmodulus of continuity and in [15] to the convolution operators (see also [12, \S \S 13,17] in thisconnection). It should be noted that the central point of the investigations in [7], [8], $[13]-[15]$

was the determination of estimates of Zygmund type for the fractional integrals $I_{a+}^{\alpha}\phi$ andthe fractional derivative $D_{a+}^{\alpha}\phi$ .

An analogue of Hardy-Littlewood’s theorem for the so-called operators with power-logarith-mic kernels

(3) $(I_{a}^{\alpha\beta} \dotplus\emptyset)(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}\log^{\beta}(\frac{\gamma}{x-t})\phi(t)dt$ ,

for $-\infty<a<b<\infty,$ $\alpha>0,$ $\gamma>b-a$ and a natural number $\beta$ , in the spaces $H_{0}^{\lambda}([a, b])$

and $H_{0^{\lambda}}([a, b];\rho)$ with the weight (2) were obtained in [5] and [6], respectively. In [6] it wasalso shown that the operator $I_{a}^{\alpha\beta}\dotplus$ with natural $\beta$ implements an isomorphism between thespaces $H_{0}^{\lambda}([a, b];\rho)$

.and $H_{0}^{\lambda+\alpha,\beta}([a, b];\rho)$ . In [12, \S 21] the former results were extended to the

operator (3) with a non-negative $\beta$ .This paper is devoted to obtain the Zygmund type estimates for the integral (3) and for its

inversion, and to apply such results to the investigation of mapping properties of the operator$I_{a}^{a\beta}\dotplus$ in the generalized H\"older spaces $H_{0}^{\omega}([a, b])$ . Section 2 contains preliminary information.Section 3 deals with the proving the Zygmund type estimate for the integral $I_{a}^{\alpha\beta}\dotplus\phi$ given in(3). Section 4 is devoted to obtain such an estimate for the integral $(I_{0}^{\alpha}\dotplus^{1})^{-1}f$ , where $(I_{a}^{\alpha}\dotplus^{1})^{-1}$

is the operator inverse to $I_{a}^{\alpha}\dotplus^{1}\cdot$ In Section 5 we apply these results to give conditions for theoperator $I_{a}^{\alpha\beta}\dotplus$ to map from the space $H_{0}^{\omega}$ into $H_{0}^{\omega_{\alpha,\beta}},$ $\omega_{\alpha)\beta}(h)=\omega(h)t^{a}\log^{\beta}(\gamma/h)$ , and to bean isomorphism of these spaces.

2. Preliminaries

Let $[a, b]$ be a finite interval of the real axis, a function $f$ be given on $[a, b]$ , and

(4) $\omega(f, h)=\sup$ $\sup$ $|f(x+t)-f(x)|$$0<t<hx,x+t\in[a,b]$

be the modulus of continuity of $f$ . Let $\omega(h)$ be a continuous and almost increasing functionon $[0, b-a]$ such that $\omega(0)=0$ . We denote by $H^{\omega}=H^{\omega}([a, b])$ the space of functions $f(x)$

with the finite norm

(5) $||f||_{H^{\omega}}=||f||_{C([a,b])}+ \sup_{h>0}\frac{\omega(f,h)}{\omega(h)}$ .

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We also denote by $H_{0}^{\omega}=H_{0}^{\omega}([a, b])$ a subspace of $H^{\omega}=H^{\omega}([a, b])$ :

(6) $H_{0}^{\omega}=H_{0}^{\omega}([a, b])=\{f\in H^{\omega} : f(a)=0\}$ ,

and define the norm by$||f||_{H_{0}^{u}}=||f||_{H^{w}}$ .

In particular, if $\omega(h)=h^{\lambda}$ , then $H^{\omega}=H^{\lambda}$ and $H_{0}^{\omega}=H_{0}^{\lambda}$ are the spaces of usual H\"olderian

functions (see, e.g. [12, \S 1.1]).For $\delta\geqq 0,$ $\nu\geqq 0$ , we say that

(7) $\omega\in\Phi_{\nu}^{\delta}$ ,

if the function $\omega(t)$ satisfy the conditions

(8) $\int_{0}^{t}(\frac{t}{\xi})^{\delta}\omega(\xi)\frac{d\xi}{\xi}\leqq c\omega(t)$ .and

(9) $\int_{t}^{b-a}(\frac{t}{\xi})^{\nu}\omega(\xi)\frac{d\xi}{\xi}\leqq c\omega(t)$

with a constant $c\geq 0$ . $\Phi_{\nu}^{\delta}$ is the subspace of the Bari-Stechkin class $\Phi_{\nu}$ (see [2]). Note thatthe class $\Phi_{\nu}^{\delta}$ is empty if $\delta\geqq\nu$ . Therefore we assume that $0<\delta<\nu$ .

Let $D_{a+}^{a}\phi$ be the Riemann-Liouville fractional derivative of order $\alpha$ with $0<\alpha<1$ :

(10) $(D_{a+}^{\alpha} \phi)(x)=\frac{1d}{\Gamma(1-\alpha)dx}\int_{a}^{x}(x-t)^{-\alpha}\phi(t)dt$, $0<\alpha<1$ .

The following assertions are true:

Theorem A. [12, Theorem 13.15] Let $\phi(x)$ be a contin$uo$us function on $[a, b]$ and $\phi(a)=0$ .If $0<\alpha<1$ , then the Zygmund type estimate

(11) $\omega(I_{a+}^{\alpha}\phi, h)\leqq ch\int_{h}^{b-a}\frac{\omega(\phi,t)}{t^{2-\alpha}}dt$ , $c>0$ ,

holds for th$e$ fractional integral $I_{a+}^{\alpha}\phi$ .

Theorem B. [12, Theorem 13.16] Let $\phi(x)$ be a continuous function on $[a, b]$ and $\phi(a)=0$ .If $0<\alpha<1$ , then th$e$ Zygmund type estimate

(12) $\omega(D_{a+}^{a}\phi, h)\leqq c\int_{0}^{h}\frac{\omega(\phi,t)}{t^{1+a}}dt$ , $c>0$ ,

holds for the fractional derivative $D_{a+}^{\alpha}\phi$ .

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Theorem C. [12, Theorem 13.17] Let $0<\alpha<1$ and $\omega(t)\in\Phi_{1-\alpha}^{0}$ . Then the operator$I_{a+}^{\alpha}$ maps $H_{0}^{\omega}$ isomorphically onto $H_{0}^{\omega_{\alpha}},$ $\omega_{a}(t)=t^{\alpha}\omega(t)$ .

It is known [10] (see also [12, \S 34.2]) that the following characterization and inversion ofthe operator $I_{0\dotplus}^{a\beta}$ given in (3) for $\beta=1$ hold valid in terms of the different construction ofMarchaud type via the special Volterra function

(13) $\mu_{a}(x)=-\int_{0}^{\infty}\frac{x^{t-\alpha}}{\Gamma(t-\alpha+1)}e^{t\psi(t)}dt$ ,

where $\alpha$ is any complex number and $\psi(z)=\Gamma’(z)/\Gamma(z)$ .

Theorem D. [12, Theorem 34.1] For a function $f\in L_{p}(a, b)(-\infty<a<b<\infty)$ , tobe representable in the form $f=I_{a}^{\alpha}\dotplus^{1}\phi(0<\alpha<1)$ with $\phi\in L_{p}(a, b)$ , it is $n$ecessary when$1<p<\infty$ and sufficient when $1\leqq p<\infty$ that the limit

(14) $(Bf)(x)= \lim_{\epsilonarrow 0}\int_{a}^{x-\epsilon}[f(x)-f(t)]\mu_{\alpha}’(x-t)dt$

exists in $L_{p}(a, b)$ (we suppose that $f(x)=0$ outside of the interval $[a,$ $b]$ ). If this conditionis satisfied, then the function $\phi(x)$ is given by

(15) $\phi(x)=\mu_{\alpha}(x-a)f(x)-(Bf)(x)$ .

From the properties of the Volterra function (13) we obtain the behaviour of $\mu_{\alpha}(x)$ andits derivative $\mu_{\alpha}’(x)$ , as $|x|arrow 0$ (see [1, \S 18.3] and [12, \S 32.1]),

(16) $\mu_{a}(x)=\frac{x^{-a}}{\Gamma(1-\alpha)\log x}[1+O(1)]$,

(17) $\mu_{\alpha}’(x)=-\frac{\alpha x^{-\alpha-l}}{\Gamma(1-\alpha)\log x}[1+O(1)]$ .

In what follows, we shall denote by $c,$ $c_{1},$ $c_{2}$ , etc. the different positive constants, which donot depend on $x$ , and suppose that all integrals will be convergent.

3. Zygmund type estimate for the integral with power-logarithmic kernel

Let a function $\phi$ be given on a finite interval $[a, b],$ $\omega(\phi, h)$ be the modulus of continuityof $\phi$ defined in (4) and $I_{a}^{\alpha\beta}\dotplus\emptyset$ be the integral (3). The following analogy of Theorem A is true:

Theorem 1. Let $\phi(x)$ be a continuous function on $[a, b]$ with $\phi(a)=0$ and $\gamma>b-a$ .Then the Zygmund type estimate

(18) $\omega(I_{a}^{a\beta}\dotplus\phi, h)\leqq c\log^{\beta}(\frac{\gamma}{h})[h^{\alpha}\omega(\phi, h)+h\int_{h}^{b-a}\frac{\omega(\phi,t)}{t^{2-\alpha}}dt]$

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holds with $0<\alpha<1$ and $\beta>0$ for the integral $I_{a}^{\alpha\beta}\dotplus\phi$ .Proof. By (3) and the hypothesis of the theorem we have

$(I_{a}^{\alpha\beta} \dotplus\phi)(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}\log^{\beta}(\frac{\gamma}{x-t}I[\phi(t)-\phi(a)]dt$ .

We denote

(19) $g(x)=\phi(x)-\phi(a),$ $\psi(x)=\int_{a}^{x}(x-t)^{\alpha-1}\log^{\beta}(\frac{\gamma}{x-t})g(t)dt$

and note that

(20) $|g(x)-g(y)|\leqq\omega(\phi, |x-y|)$ .

Let $h>0$ . For any $x,$ $x+h\in[a, b]$ we have

(21) $\psi(x+h)-\psi(x)=\int_{-h}^{x-a}\frac{g(x-t)}{(t+h)^{1-\alpha}}\log^{\beta}(\frac{\gamma}{t+h})dt-\int_{0}^{x-a}\frac{g(x-t)}{t^{1-\alpha}}\log^{\beta}(\frac{\gamma}{t})dt$

$= \int_{-h}^{0}\frac{g(x-t)-g(x)}{(t+h)^{1-\alpha}}\log^{\beta}(\frac{\gamma}{t+h})dt$

$+ \int_{0}^{x-a}[\frac{\log^{\beta}(\gamma/(t+h))}{(t+h)^{1-\alpha}}-\frac{\log^{\beta}(\gamma/t)}{t^{1-\alpha}}][g(xarrow t)-g(x)]dt$

$+g(x)[ \int_{-h}^{x-a}\frac{\log^{\beta}(\gamma/(t+h))}{(t+h)^{1-\alpha}}dt-\int_{0}^{x-a}\frac{\log^{\beta}(\gamma/t)}{t^{1-\alpha}}]dt$

$\equiv I_{1}+I_{2}+I_{3}$ .

Using (20) and making the change of variable $t=h\tau$ , we estimate $I_{1}$ :

(22) $|I_{1}| \leqq\int_{0}^{h}\frac{\omega(\phi,t)}{(h-t)^{1-\alpha}}\log^{\beta}(\frac{\gamma}{h-t})$ 協

$=h^{\alpha} \int_{0}^{1}\frac{\omega(\phi,h\tau)}{(1-\tau)^{1-a}}(\log(\frac{\gamma}{h})+\log(\frac{1}{1-\tau}))^{\beta}d\tau$

$\leqq ch^{a}\log^{\beta}(\frac{\gamma}{h})\int_{0}^{1}\frac{\omega(\phi,h\tau)}{(1-\tau)^{1-\alpha}}d\tau+ch^{\alpha}\int_{0}^{1}\frac{\omega(\phi,h\tau)}{(1-\tau)^{1-\alpha}}\log^{\beta}(\frac{1}{1-\tau})d\tau$

$\leqq c_{1}h^{\alpha}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)+c_{2}h^{\alpha}\omega(\phi, h)$

$\leqq c_{3}h^{\alpha}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)$.

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For $I_{2}$ we have

$|I_{2}| \leqq\int_{0}^{x-a}\omega(\phi,t)|(t+h)^{\alpha-1}\log^{\beta}(\frac{\gamma}{t+h})-t^{a-1}\log^{\beta}(\frac{\gamma}{t})|dt$

$=h^{\alpha} \int_{0}^{(x-a)/h}\omega(\phi, h\tau)|(\tau+1)^{\alpha-1}\log^{\beta}(\frac{\gamma}{(\tau+1)h})-\tau^{a-1}\log^{\beta}(\frac{\gamma}{h\tau})|d\tau$ .

If $x-a\leqq h$ , then

(23) $|I_{2}| \leqq h^{\alpha}\int 0^{1}\omega(\phi, h\tau)[(\tau+1)^{a-1}|\log^{\beta}(\frac{\gamma}{(\tau+1)h})|+\tau^{\alpha-1}|\log^{\beta}(\frac{\gamma}{h\tau})|]d\tau$

$\leqq c_{4}h^{\alpha}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)+c_{5}h^{\alpha}\omega(\phi, h)$

$\leqq c_{6}h^{a}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)$ .

If $x-a\geqq h$ , then by applying the mean value theorem, we obtain

(24) $|I_{2}| \leqq h^{a}(\int_{0}^{1}+\int_{1}^{(x-a)/h})\omega(\phi, h\tau)|(\tau+1)^{a-1}\log^{\beta}(\frac{\gamma}{(\tau+1)h})-\tau^{\alpha-1}\log^{\beta}(\frac{\gamma}{h\tau}I|d\tau$

$\leqq h^{\alpha}[c_{7}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)+c_{8}h\int_{1}^{(x-a)/h}\omega(\phi, h\tau)\tau^{\alpha-2}\log^{\beta}(\frac{\gamma}{h\tau})d\tau]$

$\leqq h^{\alpha}[c_{7}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)+c_{8}h^{2-\alpha}\int_{h}^{b-a}\omega(\phi, t)t^{a-2}\log^{\beta}(\frac{\gamma}{t})dt]$

$\leqq h^{\alpha}[c_{7}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)+c_{8}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)(b-a-h)]$

$\leqq c_{9}h^{\alpha}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)$.

Finally we estimate $I_{3}$ :

$|I_{3}| \leqq\omega(\phi, x-a)|\int_{0}^{x-a+h}t^{\alpha-1}\log^{\beta}(\frac{\gamma}{t})dt-\int_{0}^{x-a}t^{\alpha-1}\log^{\beta}(\frac{\gamma}{t})dt|$.

If $x-a\leqq h$ , then we have

(25) $|I_{3}| \leqq\omega(\phi, x-a)[(x-a+h)^{\alpha}\int_{0}^{1}\tau^{a-1}\log^{\beta}(\frac{\gamma}{(x-a+h)\tau})d\tau$

$+(x-a)^{\alpha} \int_{0}^{1}\tau^{a-1}\log^{\beta}(\frac{\gamma}{(x-a)\tau})d\tau]$

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$\leqq\omega(\phi, x-a)[c_{10}(x-a+h)^{\alpha}\log^{\beta}(\frac{\gamma}{x-a+h})+c_{11}(x-a+h)^{a}$

$+c_{12}(x-a)^{\alpha} \log^{\beta}(\frac{\gamma}{x-a})+c_{13}(x-a)^{\alpha}]$

$\leqq\omega(\phi, h)[c_{14}h^{\alpha}\log^{\beta}(\frac{\gamma}{h})+c_{15}h^{a}]$

$\leqq c_{16}h^{\alpha}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, h)$ .

If $x-a\geqq h$ , then

$|I_{3}| \leqq\omega(\phi, x-a)|\int_{x-a}^{x-a+h}t^{\alpha-1}\log^{\beta}(\frac{\gamma}{t})dt|$

$\leqq\omega(\phi, x-a)\log^{\beta}(\frac{\gamma}{x-a})|(x-a+h)^{a}-(x-a)^{a}|$

$\leqq ch(x-a)^{\alpha-1}\log^{\beta}(\frac{\gamma}{h})\omega(\phi, x-a)$ .

From this, applying the estimate

$(x-a)^{\alpha-1} \omega(\phi, x-a)\leqq c\int_{h}^{b-a}\omega(\phi,t)t^{a-2}dt$,

(see [7] and [12, \S 13.6]), we obtain

(26) $|I_{3}| \leqq ch\log^{\beta}(\frac{\gamma}{h}I\int_{h}^{b-a}\omega(\phi,t)t^{a-2}dt$.

Substituting these estimates (22)-(26) into (21) and taking (19) and (4) into account, wearrive at the estimate (18) which completes the proof of the theorem.

Remark 1. In [15] for the convolution integral

$(K \phi)(x)=\int_{0}^{x}k(x-t)\phi(t)dt$

with a positive kernel $k(u)$ the estimate

$\omega(\rho K\phi, h)\leqq c\int_{h}^{b-a}\frac{\omega(\phi,t)}{t}dt$

was proved under the assumption that $t^{-a}\rho(t)(0<\alpha<1)$ is a non-decreasing functionalmost everywhere on $[0, b-a]$ .

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4. Zygmund type estimate for the integral inverse to the integral with power-logarithmic kernel

Let $(I_{a}^{a\beta}\dotplus)^{-1}$ be the operator inverse to the operator $I_{a}^{\alpha\beta}\dotplus$ given in (3). It is known (see[10], [12, \S 34.2] and Theorem 2.4) that, when $\beta=1,$ $(I_{a}^{\alpha\beta}\dotplus)^{-1}f$ has the form

(27) $(I_{a}^{\alpha} \dotplus^{1})^{-1}f(x)=\mu_{a}(x-a)f(x)-\int_{a}^{x}[f(x)-f(t)]\mu_{\alpha}’(x-t)dt$,

where $\mu_{\alpha}(x)$ is the special Volterra function given in (13) and $\mu_{\alpha}’(x)$ is its derivative. Thefollowing analogy of Theorem $B$ is true.

Theorem 2. Let $f(x)$ be a continuous function on $[a, b]$ and $f(a)=0$ . Then the Zygmundtype estimate

(28) $\omega((I_{a}^{a}\dotplus^{1})^{-1}f, h)\leqq c_{1}\int_{0}^{h}\omega(f,t)|\mu_{\alpha}’(t)|dt+\omega(f, h)[c_{2}|\mu_{\alpha}(h)|+c_{3}\int_{h}^{b-a}|\mu_{a}’(t)|dt]$

holds for the function $(I_{a}^{a}\dotplus^{1})^{-1}f(x)$ given in (27).Proof. Let $h>0,$ $x,$ $x+h\in[a, b]$ ,

(29) $\phi(x)\equiv(I_{a+^{1}}^{\alpha 1})^{-1}f(x)=\mu_{\alpha}(x-a)f(x)-\int_{a}^{x}[f(x)-f(t)]\mu_{\alpha}’(x-t)dt\equiv F(x)-B(x)$ .

At first we estimate $\omega(B, h)$ . We have

(30) $B(x+h)-B(x)= \int_{0}^{x+h-a}[f(x+h)-f(x+h-t)]\mu_{\alpha}’(t)dt$

$- \int_{0}^{x-a}[f(x)-f(x-t)]\mu_{\alpha}’(t)dt$

$= \int_{0}^{x-a}[f(x+h)-f(x+h-t)-f(x)+f(x-t)]\mu_{a}’(t)dt$

$+ \int_{x-a}^{x+h-a}[f(x+h)-f(x+h-t)]\mu_{\alpha}’(t)dt$

$\equiv B_{1}+B_{2}$ .

We first estimate $B_{1}$ . If $x-a\leqq h$ , then

(31) $|B_{1}| \leqq 2\int_{0}^{x-a}\omega(f,t)|\mu_{\alpha}’(t)|dt\leqq 2\int_{0}^{h}\omega(f, t)|\mu_{\alpha}’(t)|dt$.

When $x-a\geqq h$ , we have

(32) $|B_{2}| \leqq 2\int_{0}^{h}\omega(f,t)|\mu_{a}’(t)|dt+2\omega(f, h)\int_{h}^{x-a}|\mu_{\alpha}’(t)|dt$

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$\leqq 2\int_{0}^{h}\omega(f, t)|\mu_{\alpha}’(t)|dt+2\omega(f, h)\int_{h}^{b-a}|\mu_{\alpha}’(t)|dt$ .

As far as $B_{2}$ is concerned, for $x-a\leqq h$ by using the properties of the moduli of continuity(see, e.g. [2, Chapter II, \S 1]), we obtain

(33) $|B_{2}| \leqq\int_{x-a}^{x-a+h}\omega(f,t)|\mu_{a}’(t)|dt$

$\leqq\int_{0}^{2h}\omega(f, t)|\mu_{\alpha}’(t)|dt\leqq c\int_{0}^{h}\omega(f,t)|\mu_{\alpha}’(t)|dt$.

If $x-a\geqq h$ , then making the change of variable $t=\tau+x-a$ and applying the propertiesof the moduli of continuity again, we find

(34) $|B_{2}| \leqq\int_{0}^{h}\omega(f, x-a+\tau)|\mu_{\alpha}’(x-a+\tau)|d\tau$

$\leqq c_{1}\int_{0}^{h}\omega(f, t)|\mu_{\alpha}’(t)|dt$ .

Substituting (31)-(34) into (30) and taking (4) into account we obtain the estimate

(35) $\omega(B, h)\leqq c_{2}\int_{0}^{h}\omega(f,t)|\mu_{a}’(t)|dt+c_{3}\omega(f, h)\int_{h}^{b-a}|\mu_{a}’(t)|dt$ .

Now we estimate $\omega(F, h)$ . We have

(36) $F(x+h)-F(x)=f(x)[\mu_{\alpha}(x+h-a)-\mu_{\alpha}(x-a)]+\mu_{\alpha}(x+h-a)[f(x+h)-f(x)]\equiv F_{1}+F_{2}$.

For $F_{1}$ we have

$|F_{1}|=|f(x) \int_{x-a}^{x-a+h}\mu_{\alpha}’(t)dt|\leqq\omega(f, x-a)\int_{x-a}^{x-a+h}|\mu_{\alpha}’(t)|dt\leqq\int_{x-a}^{x-a+h}\omega(f,t)|\mu_{\alpha}’(t)|dt$ .

From this by arguments similar to the above for (33), we obtain

(37) $|F_{1}| \leqq c_{4}\int_{0}^{h}\omega(f, t)|\mu_{a}’(t)|dt$ .

Finally we estimate $F_{2}$ :

(38) $|F_{2}|\leqq\omega(f, h)[|\mu_{\alpha}(x+h-a)-\mu_{\alpha}(h)|+|\mu_{\alpha}(h)|]$

$\leqq\omega(f, h)[\int_{h}^{x-a+h}|\mu_{\alpha}’(t)|dt+|\mu_{\alpha}(h)|]$

$\leqq\omega(f, h)[\int_{h}^{b-a}|\mu_{a}’(t)|dt+|\mu_{a}(h)|]$ .

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Substituting (37), (38) into (36) and taking (4) into $ac$count we arrive at the estimate

(39) $\omega(F, h)\leqq c_{4}\int_{0}^{h}\omega(f,t)|\mu_{\alpha}’(t)|dt+\omega(f, h)[\int_{h}^{b-a}|\mu_{\alpha}’(t)|dt+|\mu_{\alpha}(h)|]$ .

According to (29) from (35) and (39) (after re-denoting the constants) we obtain the esti-mate (28), which completes the proof of the theorem.

5. Mapping properties and an isomorphism implemented by operators withpower-logarithmic kernels

Let $I_{a}^{\alpha\beta}\dotplus$ be the operator (3) and $H_{0}^{\omega}$ be the generalized H\"older space (6). Mapping prop-erty of $I_{a}^{a\beta}\dotplus$ in $H_{0}^{\omega}$ is characterized by the following statement.

Theorem 3. Let $0<\alpha<1,$ $\beta\geqq 0$ , a function $\omega(t)$ be continuous and almost increasingon $[0, b-a]$ with $\omega(0)=0$ and

(40) $\int_{h}^{b-a}\frac{\omega(t)}{t^{2-a}}dt\leqq c\frac{\omega(h)}{h^{1-\alpha}}$.

Then the operator $I_{a}^{a\beta}\dotplus$ maps the generalized Holder space $H_{0}^{\omega}$ boundedly into the space$H_{0}^{\omega_{\alpha,\beta}}$ with the characteristic $\omega_{\alpha,\beta}(t)=\omega(t)t^{a}\log^{\beta}(\gamma/t)$ .

Proof. When $\beta=0$ , this theorem was proved in [7] (see also [12, \S 13.6]). We considerthe case $\beta>0$ . Let

(41) $\psi(x)=(I_{a}^{\alpha\beta}\dotplus\phi)(x)$ ,

where $\phi(x)\in H_{0}^{\omega}=H_{0}^{\omega}([a, b])$ . Then according to Theorem 1 the Zygmund type estimate(18) holds for the integral (41). Applying this estimate and the condition (40) we have

(42) $\sup_{0<h\leq b-a}\frac{\omega(\psi,h)}{\omega(h)h^{a}\log^{\beta}(\gamma/h)}\leqq c[\frac{\omega(\phi,h)}{\omega(h)}+h^{1-\alpha}\int_{h}^{b-a}\frac{\omega(\phi,t)}{t^{2-a}}dt]\leqq c||\phi||_{H_{0^{y}}}$ .

The equality $\psi(a)=0$ follows from the definition (3) of the operator $I_{a}^{\alpha\beta}\dotplus$ with power-logarithmic kernel. Further, we have

(43) $|| \psi||_{C([a,b])}=\max_{a\leq x\leq b}|\frac{1}{\Gamma(\alpha)}\int_{a}^{x}(x-t)^{\alpha-1}\log^{\beta}(\frac{\gamma}{x-t})\phi(t)dt|\leqq c_{1}||\phi||_{C([a,b])}$ .

From (41)-(43) and the definition (6) of the space $H_{0}^{\omega}$ we obtain

$\Vert I_{a}^{\alpha\beta}\dotplus\emptyset\Vert_{H_{0}^{u_{\alpha,\beta}}}=||\psi||_{H_{0}^{\omega_{\alpha,\beta}}}\leqq||\phi||_{H_{0}}$. .

The theorem is proved.

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Corollary 1. Let $0<\alpha<1,$ $\beta\geqq 0,$ $\lambda>0$ and $\lambda+\alpha<1$ , then the operator $I_{a}^{\alpha\beta}\dotplus$ maps$H_{0}^{\lambda}$ boundedly into $H_{0}^{\lambda+\alpha,\beta}$ .

Remark 2. Corollary 1 was obtained by direct estimates in [5] (see also [1?, \S 21]).

Now we consider the mapping property of the operator $(I_{a}^{a}\dotplus^{1})^{-1}$ given in (27) on $H_{0}^{\omega_{\alpha,1}}$

with $\omega_{\alpha,1}(t)=\omega(t)t^{\alpha}|\log(t)|$ .

Theorem 4. Let $0<\alpha<1$ , a function $\omega(t)$ be continuous and almost increasing on$[0, b-a]$ with $\omega(0)=0$ and

(44) $\int_{0}^{h}\frac{\omega(t)}{t}dt\leqq c\omega(h)$ .

Then the operator $(I_{a}^{a}\dotplus^{1})^{-1}$ maps the generalized Holder space $H_{0}^{\omega_{\alpha,1}}$ with the characteristic$\omega_{\alpha,1}(t)=\omega(t)t^{\alpha}|\log(t)|$ boundedly into the space $H_{0}^{\omega}$ .

Proof. Let $f(x)\in H_{0}^{\omega_{\alpha,1}}=H_{0}^{\omega_{\alpha,1}}([a, b])$ , then in view of (27) we have

(45) $g(x) \equiv(I_{a}^{\alpha}\dotplus^{1})^{-1}f(x)=\mu_{\alpha}(x-a)f(x)-\int_{a}^{x}[f(x)-f(t)]\mu_{\alpha}’(x-t)dt$.

We show that

(46) $\sup_{0<h\leq b-a}\frac{\omega(g,h)}{\omega(h)}\leqq c<\infty$.

Applying the Zygmund type estimate (28), the relations (16) and (17) for the special Volterrafunction (13) and its derivative, and also the condition (44), we have

$\frac{\omega(g,h)}{\omega(h)}\leqq\frac{1}{\omega(h)}[c_{1}\int_{0}^{h}\omega(f,t)|\mu_{a}’(t)|dt+\omega(f, h)(c_{2}|\mu_{a}(h)|+c_{3}\int_{h}^{b-a}|\mu_{\alpha}’(t)|dt)]$

$\leqq||f||_{H_{0}^{u_{\alpha,1}}}-\frac{c_{1}}{\omega(h)}\int_{0}^{h}\omega(t)t^{\alpha}|\log(t)||\mu_{\alpha}’(t)|dt$

$+h^{\alpha}| \log(h)|(c_{2}|\mu_{\alpha}(h)|+c_{3}\int_{h}^{b-a}|\mu_{\alpha}’(t)|dt)]$

$\leqq||f||_{H_{0}^{\omega_{\alpha.1}}}[\frac{c_{4}}{\omega(h)}\int_{0}^{h}\frac{\omega(t)}{t}+c_{6}+c_{6}h^{\alpha}|\log(h)|\int_{h}^{b-a}\frac{dt}{t^{a+1}|1og(t)|}]$

$\leqq||f||_{H_{0}^{\alpha,1}}\cdot[c_{7}+c_{8}|\log(h)|\int_{1}^{(b-a)l^{h}}\frac{d\tau}{\tau^{\alpha+1}|\log(h\tau)|}]$

$\leqq c_{9}||f||_{H_{0}^{\omega_{\alpha,1}}}$ .

From here we obtain the estimate of the form (46):

(47) $\sup_{0<h\leq b-a}\frac{\omega(g,h)}{\omega(h)}\leqq c_{9}||f||_{H_{0}^{\alpha,1}}\cdot$ .

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Now we estimate $||g||_{C([a,b])}$ . We have

$||g||_{C([a,b])} \leqq\sup_{a<x\leq b}[|\mu_{\alpha}(x-a)|\omega(f)x-a)+\int_{0}^{x-a}|\mu_{a}’(t)|\omega(f, t)dt]$

$\leqq c||f||_{H_{0}^{w_{\alpha,1}}}[(x-a)^{a}|\log(x-a)|\omega(x-a)|\mu_{a}(x-a)|$

$+ \int_{0}^{x-a}t^{a}|\log(t)|\omega(t)|\mu_{a}’(t)|dt]$

$\leqq||f||_{H_{0}^{u_{\alpha,1}}}[c_{10}\omega(x-a)+c_{11}\int_{0}^{x-a}\frac{\omega(t)}{t}dt]$

$\leqq c_{12}||f||_{H_{0}^{u_{\alpha,1}}}\omega(x-a)$ .

From here we have$||g||_{C([a,b])}\leqq c_{12}||f||_{H_{0}^{\omega_{\alpha,1}}}$ ,

and tahng (46) and (47) into account, we finally arrive at the estimate

$||g||_{H_{0}}$. $\leqq c||f||_{H_{0}^{\alpha,1}}\cdot$ .

The condition $g(a)=0$ follows directly from (45) if we take the relations (16), (17) and (44)into account. This completes the proof of this theorem.

If $X$ and $Y$ are Banach spaces and $T$ is an operator, we.denote by $T$ : $X-Y$ theimbedding with the properties

(i) if $f\in X$ , then $Tf\in Y$ ;

(ii) $||Tf||_{Y}\leqq c||f||_{X}$ .

Thus, in Theorems 3 and 4 we have proved the following imbeddings:

(48) $I_{a}^{a\beta}\dotplus$ : $H_{0}^{\omega}\mapsto H_{0}^{\omega_{\alpha},\rho},$ $0<\alpha<1,$ $\beta\geqq 0$ ,

and

(49) . $(I_{a+}^{a1}))^{-1}$ : $H_{0}^{\omega_{\alpha,1}} H_{0}^{\omega},$ $0<\alpha<1-$ .

Thus we obtain the analogy of Theorem $C$ about an isomorphism of the generalized H\"older

spaces $H_{0}^{\omega}$ and $H_{0}^{\omega_{\alpha,1}}$ implemented by the operator $I_{a}^{\alpha}\dotplus^{1}$ with the power-logarithmic kernel.

Theorem 5. Let $0<\alpha<1,$ $\beta\geqq 0$ an$d\omega(t)\in\Phi_{1-\alpha}^{0}$ , where $\Phi_{1-\alpha}^{0}$ is the $sp$ace definedin (7). Then the operator $I_{a}^{\alpha}\dotplus^{1}$ maps the space $H_{0}^{\omega}$ isomorphically onto the space $H_{0}^{\omega_{\alpha,1}}$ withthe characteristic $\omega_{\alpha,1}(t)=\omega(t)t^{\alpha}|\log(t)|$ .

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Proof. To show that the assertion of this theorem follows from (48) and (49) we haveto prove that any function $f\in H_{0}^{\omega_{\alpha.1}}$ is representable by the integral (3) $f=I_{a}^{\alpha}\dotplus^{1}\phi$ witha function $\phi\in H_{0}^{\omega}$ . For this we use the criterion of representability of a function $f$ viathe power-logarithmic integral $f=I_{a}^{a}\dotplus^{1}\emptyset$ of a function $\phi\in L_{p}(a, b)$ given in Theorem D.We verify that the conditions of Theorem $D$ hold for a function $f\in H_{0}^{\omega_{\alpha,1}}$ . The condition$f\in L_{p}(a, b)$ is valid because

$|f(x)|\leqq\omega(x-a)(x-a)^{\alpha}|\log(x-a)|||f||_{H_{0}^{\alpha,1}}\cdot\leqq c$ .

We verify the convergence in $L_{p}(a, b)$ of the functions as $\epsilonarrow 0$

(50) $\psi_{\epsilon}(x)=\{\begin{array}{l}\int_{a}^{x-a}[f(x)-f(t)]\mu_{\alpha}’(x-t)dtifa+\epsilon<x<b0ifa<x<a+\epsilon\end{array}$

It is sufficient to show that the sequence $\psi_{\epsilon}(x)$ is fundamental in the space $L_{p}(a, b)$ . Wesuppose that $\epsilon_{1}<\epsilon_{2}$ and put $x>a+\epsilon_{2}$

$\psi_{\epsilon_{1}}(x)-\psi_{\epsilon_{2}}(x)=\int_{a}^{x-\epsilon_{1}}[f(x)-f(t)]\mu_{\alpha}’(x-t)dt-\int_{a}^{x-\epsilon_{2}}[f(x)-f(t)]\mu_{\alpha}’(x-t)dt$

$= \int_{\epsilon_{1}}^{\epsilon_{2}}[f(x)-f(x-t)]\mu_{\alpha}’(t)dt$ .

Since $\omega(f,t)\leqq c\omega(t)t^{\alpha}|\log(t)|$ and by (17) $|\mu_{a}’(t)|\leqq ct^{-a-1}|\log(t)|^{-1}$ , then

$| \psi_{\epsilon_{1}}(x)-\psi_{\epsilon_{2}}(x)|\leqq c\int_{\epsilon_{1}}^{\epsilon_{2}}\frac{\omega(t)}{t}dtarrow 0$ $(\epsilon_{2}arrow 0)$ .

The cases $x<a+\epsilon_{1}$ and $a+\epsilon_{1}<x<a+\epsilon_{2}$ are considered similarly. Thus, the sequence$\psi_{\epsilon}(x)$ in (50) is fundamental in the norm of the space $C([a, b])$ , and hence also in the normof $L_{p}(a, b)$ . According to (45) and (49) the function $\phi$ in the repesentation

$f=I_{a}^{\alpha}\dotplus^{1}\phi,$ $\phi\in L_{p}(a, b),$ $1<p<\infty$ ,

belongs to $H_{0}^{\omega}$ . This completes the proof of the theorem.

Corollary 2. If $0<\alpha<1,$ $\lambda>0$ and $\lambda+\alpha<1$ , then the operator $I_{a}^{\alpha}\dotplus^{1}$ maps the space$H_{0}^{\lambda}$ isomorphically onto the space $H_{0}^{\lambda+a,1}$ .

Remark 3. In [6] the statement more general than Corollary 2 was proved giving theconditions for the operator $I_{a}^{\alpha\beta}\dotplus,$ $\beta=1,2,$ $\ldots$ , to be an isomorphism between the generalizedweighted H\"older spaces $H_{0}^{\lambda}([a, b];\rho)$ and $H_{0}^{\lambda+a,\beta}([a, b];\rho)$ , where $\rho$ is the weight (2).

100

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